Oscillation signals such as seismic waves have been analyzed, using an autoregressive model. For this analysis, dumped oscillation signals are sampled at a uniform-time interval.
The Sompi method utilizing autoregressive models has been recently proposed to offer higher accuracy than the maximum entropy method (MEM) and the linear prediction method, which are also based on autoregressive methods (J. Geophys. Res. 94, 7535 (1989)). The Sompi method algebrically derives the parameters of wave elements from the time series data, sampled at a uniform-time interval. Time series data is described in a linear combination of dumped oscillation signals as equation (1), where the subscript denotes the wave element. EQU x.sub.tl =.SIGMA.A.sub.l exp (-2.pi.g.sub.l .DELTA.t.multidot.t) cos (2.pi.F.sub.l .DELTA.t.multidot.t+.phi..sub.l) (1)
The parameters include frequency (f), attenuation coefficient (g), initial phase (.phi.), and initial amplitude (A). In the above equation, t is a variable expressing time taken discretely, and .DELTA.t is the sampling interval.
If the parameters of various waves are found, it is easy to represent them in the form of a spectrum on the frequency (f) axis by substituting the parameters into a Lorentzian distribution function.
In the Sompi method data {x.sub.t } taken at a uniform-time and given by equation (1) is separated into signal component {u.sub.t } and noise component {n.sub.t }. That is, EQU x.sub.t =u.sub.t +n.sub.t ( 2)
It is now assumed that the noise component has a Gaussian distribution.
If the nature of the signal component can be completely determined, i.e., it can be described in terms of f, g, .phi., and A, then the signal component can be completely described, using an autoregressive model, as given by equation (3). ##EQU1## where a.sub.j is an autoregressive coefficient, and m is the predetermined maximum autoregressive order.
The autoregressive equation used in the linear prediction method contains noise component as given by equation (4), and spectral components are biased. ##EQU2## When the data {x.sub.t } is expressed using an autoregressive model as given by equation (3), if the fitting error is given by a statistical amount S, then this amount is given as follows. ##EQU3## where ##EQU4## and .sigma..sup.2 s the variance.
An autoregressive model at the order m can be determined by finding the autoregressive coefficients which minimize the fitting error, given by equation (5).
We now give condition (6) ##EQU5## Using Lagrange's Method of undetermined multiplier, we have EQU P.multidot.A=.lambda.A (17)
where ##EQU6## The minimum eigenvalue .lambda..sub.1 of equation (7) provides the minimum fitting error (power of noise). An eigenvector corresponding to this eigenvalue gives an autoregressive coefficient. In this way, autoregressive coefficients are derived. In the Sompi method, solutions, i.e., the frequencies f and the attenuation constants g, are derived from these autoregressive coefficients in the manner described below.
Taking the z transform of equation (3) results in EQU A(z).multidot.U(z)=0 (8)
where ##EQU7## In equation (11), .DELTA.t is the aforementioned sampling interval. In equation (8), since a signal is given by U(z), if any signal component exists, then U(z).noteq.0. Under this condition, in order that equation (8) hold at all times, the relation, A (z).noteq.0, must hold.
After substituting the autoregressive coefficients calculated from equation (7) into equation (9), the algebraic equation, A (z)=0, is solved for z. The solutions are modified using equations (11) and (12) to find spectral parameters f and g. The values of f and g are substituted into equation (1). The other characteristic parameters A and .phi. of the wave elements can be found by applying the least squares method to data x.sub.t sampled at uniform-time interval. Using these results and equation (1), the power of the spectrum is given by EQU PW.sub.l =(1/N).multidot..SIGMA.x.sub.tl.sup.2
where N is the data length.
In a system described by an autoregressive model, if the autoregressive order is set less than the number of spectral peaks contained in a spectrum in practice, then the resolution of the obtained spectrum deteriorates. Conversely, if the order is made too large, the resolution is improved but spurious spectra are contained in the results.
In many cases, autoregressive orders suitable for unknown data have been heretofore determined empirically. This is laborious to perform. In order to avoid this laborious procedure, theoretical determination of appropriate autoregressive orders has been attempted. H. Akaike has proposed so-called Akaike's information criterion, AIC (m), to determine optimum autoregressive orders. According to Akaike's theory, the value of m ch minimizes the AIC is the optimum order. EQU AIC(m)=N log .sigma..sup.2 (m)+2B 1
where m is the autoregressive order, N is the data length, B is the number of parameters, and .sigma..sup.2 (m) is the variance obtained when the autoregressive order is m.
Time series data taken from a free induction decay signal obtained from a pulse NMR spectrometer contains two components, cosine component and sine component, which are .pi./2 out-of-phase. It is the common practice to process such signals by complex Fourier transform. However, data analysis by Fourier transform cannot separate noise component from intrinsic signal component. One conceivable method of solving this problem is to analyze data obtained from a pulse NMR spectrometer by the Sompi method. In the convention Sompi method, only real parts, or the cosine components, are analyzed. Therefore, the conventional Sompi method is not adequate to analysis of data obtained by a pulse NMR spectrometer.
In analysis of observed time series data, typically the data obtained by pulse NMR spectroscopy, often contain large amounts of noise. In such a case, if above-described Akaike's information criterion, AIC (m), is used to estimate the optimum autoregressive order m at which the norm AIC (m) assumes its minimum value, then AIC may assume no locally minimum value provided that a large amount of noise is contained. This makes it difficult to select the optimum autoregressive order from plural autoregressive orders, by the Sompi method or other method.
The conventional method of displaying the results of an analysis using an autoregressive model is only to provide a graphical display of the solutions on the f-g plane. Therefore, it is not easy to compare this graphical display with a spectrum which is drawn on the f-axis as frequently done in the prior art techniques.